On the variability of case-deletion importance sampling weights in the Bayesian linear model

I consider a standard specification of the Bayesian linear model and derive necessary and sufficient conditions for the variance of the case-deletion importance sampling weights to be finite. The conditions have an intuitive interpretation in terms of familiar frequentist measures of leverage and influence and are easy to verify. I present two real data examples in which the necessary conditions fail to hold for some observations and the corresponding importance sampling estimates are highly unreliable.